Are Ashkenazi Jews decendants of Khazars? Was there a mass conversion of Khazars to Judaism?

As Juicy pointed out, this is a topic of heavy debate among scholars. I'd like to add that recent research suggests that there is some reason to believe that yes, Ashkenazi Jews are the descendants of the Khazars or at least peoples in their vicinity.

Specifically, Dr. Eran Elhaik has brought up this issue again in a paper published in 2012, where he concludes that:

We conclude that the genome of European Jews is a tapestry of ancient populations including Judaized Khazars, Greco–Roman Jews, Mesopotamian Jews, and Judeans and that their population structure was formed in the Caucasus and the banks of the Volga with roots stretching to Canaan and the banks of the Jordan.

Though the paper is being contested (largely on political grounds), it does reopen the door for the Khazar hypothesis.

I cannot comment on the genetic lineage of Ashkenazi Jews, though I will attempt to explain their ethnic origins from a historical standpoint.

Ashkenazi Jews currently comprise the majority of the world's Jewish population-(approximately 80% worldwide). This particular Jewish community has its geographical origins in the European continent, though their anthropological origins are of mixed ethnic heritage.

In the Middle Ages, there were Ashkenazi Jewish communities living in Northern Italy, Northern France, England and the Germanic Rhineland/Western Germany. Yiddish, the central language of the Ashkenazi Jews, was essentially born in the Germanic lands during the Middle Ages and was-(and is still), a combination of German and Hebrew or Germanic languages and Semitic languages related to Hebrew-(which could also include Aramaic).

After the Black Death in the 1300's CE, many Jewish communities within Northern Europe were expelled and subsequently relocated to Poland at the invitation of a Polish King-(I do not remember his name). Throughout much of the Modern era, a sizable percentage of Ashkenazi Jews had been living throughout Poland. However, during the reign of Catherine The Great-(if my memory is correct), many Polish Jews were uprooted and resettled into other parts of the Russian Empire, including Russia proper, as well neighboring lands to the West and South of Russia.

By the 19th and 20th centuries, Ashkenazi Jewish communities were primarily living in Eastern Europe-(with a sizable percentage living in Russia and Poland), though other Ashkenazi Jewish communities were living Austria, Germany and France, as well as emigrating to the United States, Canada, England and Argentina during the early 20th century.

This is the mainstream and generally accepted history of Ashkenazi Jewish migration through the centuries. The so-called Khazar theory or thesis, has been and is still, rooted in conjecture and controversy for many reasons.

First, the Khazars were Mongol-Turks from Central Asia, who traveled the Silk Road en route to what is today, the Ukraine and parts of Georgia around 800 CE. They established an empire that served as a type of regional partner for the neighboring Byzantine Empire's rule around the Black Sea region for approximately 300 years. There were some Khazars-(mostly the Nobility), who converted to Judaism during this time, though their numbers were fairly small when compared with the majority of the Khazar civilian population. Exactly what happened after the fall of the Khazarian Empire is somewhat historically mysterious. In all likelihood, the Khazars probably intermarried with local Slavic women living in the Ukraine or Russian Black Sea region. Perhaps there were inter-religious marriages between converted Khazrian Jews and Slavic (mostly Eastern rite) Christians. Again, there is little historical evidence which affirms this and is typically subjected to conjecturing and hypothesizing. Even if all or the majority of the Khazar theory were historically true, it is still historically parenthetical when compared with the more mainstream historical explanations which describe the mixed European and Semitic ethnic origins of the majority of Ashkenazi Jews, which, in all likelihood, was probably due to centuries of intermarriage between European Jewish men and European Christian women.

There is a likely correlation between the ethnic and genetic origins of Ashkenazi Jews which is primarily rooted in the 1800 year history of the European Jewish Diaspora. In other words, Ashkenazi Jews-(from a genealogically ethnic standpoint), greatly reflect the European populations they lived among for several centuries, while also retaining their original Middle Eastern Semitic origins. I would say that in genealogically ethnic terms, most Ashkezazi Jews are neither totally Semitic, nor totally European, but rather, a mixture of both groups. The Khazar theory, if true, would probably have parenthetical or little significance when compared with the historically and chronologically documented evidence of the 1800 year European Ashkenazi Jewish Diaspora.

Another important marker of lineage is culture, and in particular language which is by far the best studied cultural feature.

The language of the Ashkenazi is Yiddish. This language is actually a Germanic language from the High German branch. This culture, including the language, is thought to have evolved in what is now Germany in the 10th Century, and only later spread east. So it would be quite reasonable to assume that most of the non-Semetic roots of the Ashkenazi are in fact German.

German is in fact quite unrelated to Turkic Khazar, the language of the Khazars (and a considerable distance away). So there is really no good reason to believe there was any serious Khazar component to this culture.

This appears to back up the other genetic-based answers. The most likely origin appears to be that the culture was founded by male Jewish emigres who intermarried with local German women. The children would have learned to speak the German their mothers spoke at home, while the fathers would have insisted the religious trappings of Judaism be adhered to.

There is evidence that the Khazar nobility had converted to Judaism by the 10th century. This seems to be a response to pressure from Orthodox Christians and Muslims (surrounding them). Being monotheistic would have the advantage of making it harder for those powers to justify invasion/assimilation of the Khazars and choosing Judaism would have guaranteed them some form of independence. However most believe only the nobility converted to Judaism. Either way, the theory that Khazars migrated west and became the Ashkenazi after the dissolution of their empire doesn't have much support among scholars.

This seems to be backed by genetic studies:

A 2013 study of Ashkenazi mitochondrial DNA, from the University of Huddersfield in England, suggests that at least 80 percent of the Ashkenazi maternal lineages derive from the assimilation of mtDNAs indigenous to Europe

[Another] study states that the great majority of Ashkenazi maternal lineages were not brought from the Near East (i.e., they were non-Israelite), nor were they recruited in the Caucasus (i.e., they were non-Khazar), but instead they were assimilated within Europe.

Both citations taken from the Wikipedia page on Ashkenazi Jews.

So it seems the conversion of Khazars (if it did indeed happen en masse) had relatively little impact on the overall bloodline of Ashkenazi Jews. The studies agree that Ashkenazis form a distinct ethnic group, mostly because they haven't intermarried much for at least the last thousand years. However both studies agree that they emerged out of European bloodlines and not Near-Eastern or Caucasus. As such they are perhaps not "truly Semitic" and probably not descendants of the ancient Israelites, at least not directly or by a majority.

Some historians suggest that Hellenistic era Judaism may have been a proselytizing faith, ie: they actively attempted to convert pagans to their faith. If this is the case, Ashkenazi Jews may be descendants of early converts from southern Europe. There is a question on proselytism on SE:Judaism, in general the answers and comments seem to agree that some form of proselytizing did occur in the past.

UPDATE

After further research it seems that the studies focusing on maternal lineages (through mitochondrial DNA, which is passed unchanged from mother to child) point to the fact the majority of women that eventually formed the Ashkenazis were indigenous to Europe, while studies focusing on paternal haplotypes find stronger links to the Middle-East in 50% of Ashkenazis Levites. Although this priestly class represents only 4% of the Ashkenazi population, this may indicate that very few men from the Middle-East formed the base of the Ashkenazi priestly class. One could imagine that a handful of priests migrated to Europe and formed the core of what would become the Ashkenazi.